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Qiongling Li

Publications and source records attributed to Qiongling Li.

At least 19 recordsLinked to original sources
arXiv2026

Non-maximal quasi-isometric $\mathrm{PU}_{2,n+1}$-representations via alternating surfaces in complex pseudo-hyperbolic spaces

For every admissible non-maximal Toledo invariant $t$, we construct a locus of irreducible representations of a closed genus-$g$ surface group into $\mathrm{PU}_{2,n+1}$ with Toledo invariant $t$ whose orbit maps are quasi-isometric embeddings. These loci have real codimension $10(g-1)$ or $10(g-1)+2(n-1)$ in the character variety depending on the Toledo value. Our construction uses $4$-cyclic Higgs bundles and their correspondence with $\partial$-alternating surfaces in complex pseudo-hyperbolic spaces. The associated equivariant minimal maps into the symmetric space are bi-Lipschitz embeddings. We further construct cocompact domains of discontinuity in the Shilov boundary. Their quotients are smooth fiber bundles over the surface with fiber homeomorphic to $S^{2n-1}\times S^{2n-1}$.

math.DG↗
arXiv2026

Harmonic Maps from Punctured Riemann Surfaces to the Hyperbolic Plane with Prescribed Scherk Asymptotics

Let $X$ be a genus $g\geq 0$ compact Riemann surface that admits an antiholomorphic involution $ι$ with non empty fixed-point set, and let $D = \{p_1,\dots,p_k\}\subset \text{Fix}(ι)$. At each puncture, we prescribe a Scherk map associated to a given real polynomial quadratic differential with even degree and negative leading coefficient. Here a Scherk map is the harmonic diffeomorphism from $\mathbb{C}$ to the interior of an ideal polygon in $\mathbb{H}^2$, whose Hopf differential is the given polynomial. We construct a harmonic map \[ h:X\backslash D \to \mathbb{H}^2 \] whose asymptotic behavior at each puncture matches the prescribed Scherk map. In particular, the image of $h$ tends to an ideal polygon near each end. The resulting $h$ covers harmonic maps obtained by taking the $\mathbb{H}^2$ factors of horizontal catenoids in $\mathbb{H}^2 \times \mathbb{R}$.

math.DG↗
arXiv2026

Infinitesimal Rigidity of Cyclic Surfaces and Alternating Surfaces

We study the infinitesimal rigidity of equivariant minimal maps from the universal cover of a smooth oriented surface (possibly non-compact) into a Riemannian symmetric space, focusing on representations arising from cyclic harmonic bundles. By developing a unified Lie-theoretic framework that connects cyclic surfaces and cyclic harmonic bundles over Riemann surfaces, we prove the infinitesimal rigidity for irreducible cyclic surfaces under admissible smooth variations, including both compactly supported deformations and $L^p$-integrable variations on non-compact surfaces. As a geometric application, we introduce $n$-alternating surfaces in $\mathbb H^{p,q}$ and establish their correspondence with a special class of cyclic surfaces. This yields an infinitesimal rigidity theorem that conceptually unifies and extends known rigidity results for maximal space-like surfaces, alternating holomorphic curves, and $A$-surfaces in certain $\mathbb H^{p,q}$.

math.DG↗
arXiv2024

Harmonic metrics of generically regular nilpotent Higgs bundles over non-compact surfaces

A rank $n$ Higgs bundle $(E,θ)$ is called generically regular nilpotent if $θ^n=0$ but $θ^{n-1}\neq 0$. We show that for a generically regular nilpotent Higgs bundle, if it admits a harmonic metric, then its graded Higgs bundle admits a unique maximal harmonic metric. The proof relies on a generalization of Kalka-Yang's theorem for prescribed curvature equation over a non-compact hyperbolic surface to a coupled system. As an application, we show that the branched set of a branched minimal disk in $\mathbb{H}^3$ has to be the critical set of some holomorphic self-map of $\mathbb{D}$.

math.DG↗
arXiv2024

Isolated singularities of Toda equations and cyclic Higgs bundles

This paper is the second part of our study on the Toda equations and the cyclic Higgs bundles associated to $r$-differentials over non-compact Riemann surfaces. We classify all the solutions up to boundedness around the isolated singularity of an $r$-differential under the assumption that the $r$-differential is meromorphic or has some type of essential singularity. As a result, for example, we classify all the solutions on ${\mathbb C}$ if the $r$-differential is a finite sum of the exponential of polynomials.

math.DG↗
arXiv2024

Every closed surface of genus at least 18 is Loewner

In this paper, we obtain an improved upper bound involving the systole and area for the volume entropy of a Riemannian surface. As a result, we show that every orientable and closed Riemannian surface of genus $g\geq 18$ satisfies Loewner's systolic ratio inequality. We also show that every closed orientable and nonpositively curved Riemannnian surface of genus $g\geq 11$ satisfies Loewner's systolic ratio inequality.

math.DG↗
arXiv2023

Harmonic metrics of generically regular semisimple Higgs bundles on non-compact Riemann surfaces

We prove that a generically regular semisimple Higgs bundle equipped with a non-degenerate symmetric pairing on any Riemann surface always has a harmonic metric compatible with the pairing. We also study the classification of such compatible harmonic metrics in the case where the Riemann surface is the complement of a finite set $D$ in a compact Riemann surface. In particular, we prove the uniqueness of a compatible harmonic metric if the Higgs bundle is wild and regular semisimple at each point of $D$.

math.DG↗
arXiv2023

Higgs bundles in the Hitchin section over non-compact hyperbolic surfaces

Let $X$ be an arbitrary non-compact hyperbolic Riemann surface, that is, not $\mathbb C$ or $\mathbb C^*$. Given a tuple of holomorphic differentials $\boldsymbol q=(q_2,\cdots,q_n)$ on $X$, one can define a Higgs bundle $(\mathbb{K}_{X,n},θ(\boldsymbol q))$ in the Hitchin section. We show there exists a harmonic metric $h$ on $(\mathbb{K}_{X,n},θ(\boldsymbol q))$ satisfying (i) $h$ weakly dominates $h_X$; (ii) $h$ is compatible with the real structure. Here $h_X$ is the Hermitian metric on $\mathbb{K}_{X,n}$ induced by the conformal complete hyperbolic metric $g_X$ on $X.$ Moreover, when $q_i(i=2,\cdots,n)$ are bounded with respect to $g_X$, we show such a harmonic metric on $(\mathbb{K}_{X,n},θ(\boldsymbol q))$ satisfying (i)(ii) uniquely exists. With similar techniques, we show the existence of harmonic metrics for $SO(n,n+1)$-Higgs bundles in Collier's component and $Sp(4,\mathbb R)$-Higgs bundles in Gothen's component over $X$, under some mild assumptions.

math.DG↗
arXiv2022

Bounded differentials on unit disk and the associated geometry

For a harmonic diffeomorphism between the Poincaré disks, Wan showed the equivalence between the boundedness of the Hopf differential and the quasi-conformality. In this paper, we will generalize this result from quadratic differentials to $r$-differetials. We study the relationship between bounded holomorphic $r$-differentials and the induced curvature of the associated harmonic maps from the unit disk to the symmetric space $SL(r,\mathbb R)/SO(r)$ arising from cyclic/subcyclic harmonic Higgs bundles. Also, we show the equivalences between the boundedness of holomorphic differentials and having a negative upper bound of the induced curvature on hyperbolic affine spheres in $\mathbb{R}^3$, maximal surfaces in $\mathbb{H}^{2,n}$ and $J$-holomorphic curves in $\mathbb{H}^{4,2}$ respectively. Benoist-Hulin and Labourie-Toulisse have previously obtained some of these equivalences using different methods.

math.DG↗
arXiv2021

Projective Structures with (Quasi-)Hitchin Holonomy

In this paper we investigate the properties of the real and complex projective structures associated to Hitchin and quasi-Hitchin representations that were originally constructed using Guichard-Wienhard's theory of domains of discontinuity. We determine the topology of the underlying manifolds and we prove that some of these geometric structures are fibered in a special standard way. In order to prove these results, we give two new ways to construct these geometric structures: we construct them using gauge theory, flat bundles and Higgs bundles, and we also give a new geometric way to construct them.

math.GT↗
arXiv2020

Complete solutions of Toda equations and cyclic Higgs bundles over non-compact surfaces

On a Riemann surface with a holomorphic $r$-differential, one can naturally define a Toda equation and a cyclic Higgs bundle with a grading. A solution of the Toda equation is equivalent to a harmonic metric of the Higgs bundle for which the grading is orthogonal. Here we focus on a general non-compact Riemann surface with an $r$-differential which is not necessarily meromorphic at infinity. We introduce the notion of complete solution of the Toda equation, and we prove the existence and uniqueness of a complete solution by using techniques for both Toda equations and harmonic bundles. Moreover, we show some quantitative estimates of the complete solution.

math.DG↗
arXiv2020

Nilpotent Higgs bundles and the Hodge metric on the Calabi-Yau moduli

We study an algebraic inequality for nilpotent matrices and show some interesting geometric applications: (i) obtaining topological information for nilpotent polystable Higgs bundles over a compact Riemann surface; (ii) obtaining a sharp upper bound of the holomorphic sectional curvatures of the period domain and the Hodge metric on the Calabi-Yau moduli.

math.DG↗
arXiv2020

Domination results in $n$-Fuchsian fibers in the moduli space of Higgs bundles

In this article, we show some domination results on the Hitchin fibration, mainly focusing on the $n$-Fuchsian fibers. More precisely, we show the energy density of associated harmonic map of an $n$-Fuchsian representation dominates the ones of all other representations in the same Hitchin fiber, which implies the domination of topological invariants: translation length spectrum and entropy. As applications of the energy density domination results, we obtain the existence and uniqueness of equivariant minimal (or maximal) surfaces in certain product Riemannian (or pseudo-Riemannian) manifold. Our proof is based on establishing an algebraic inequality generalizing a GIT theorem of Ness on the nilpotent orbits to general orbits.

math.DG↗
arXiv2019

An Introduction to Higgs Bundles via Harmonic Maps

This survey studies equivariant harmonic maps arising from Higgs bundles. We explain the non-abelian Hodge correspondence and focus on the role of equivariant harmonic maps in the correspondence. With the preparation, we review current progress towards some open problems in the study of equivariant harmonic maps.

math.AG↗
arXiv2018

Harmonic maps for Hitchin representations

Let $(S,g_0)$ be a hyperbolic surface, $ρ$ be a Hitchin representation for $PSL(n,\mathbb R)$, and $f$ be the unique $ρ$-equivariant harmonic map from $(\widetilde S, \widetilde g_0)$ to the corresponding symmetric space. We show its energy density satisfies $e(f)\geq 1$ and equality holds at one point only if $e(f)\equiv 1$ and $ρ$ is the base $n$-Fuchsian representation of $(S,g_0)$. In particular, we show given a Hitchin representation $ρ$ for $PSL(n,\mathbb R)$, every $ρ$-equivariant minimal immersion $f$ from a hyperbolic plane $\mathbb H^2$ into the corresponding symmetric space $X$ is distance-increasing, i.e. $f^*(g_{X})\geq g_{\mathbb H^2}$. Equality holds at one point only if it holds everywhere and $ρ$ is an $n$-Fuchsian representation.

math.DG↗
arXiv2017

On cyclic Higgs bundles

In this paper, we derive a maximum principle for a type of elliptic systems and apply it to analyze the Hitchin equation for cyclic Higgs bundles. We show several domination results on the pullback metric of the (possibly branched) minimal immersion $f$ associated to cyclic Higgs bundles. Also, we obtain a lower and upper bound of the extrinsic curvature of the image of $f$. As an application, we give a complete picture for maximal $Sp(4,\mathbb{R})$-representations in the $2g-3$ Gothen components and the Hitchin components.

math.DG↗
arXiv2017

On the uniqueness of vortex equations and its geometric applications

We study the uniqueness of a vortex equation involving an entire function on the complex plane. As geometric applications, we show that there is a unique harmonic map $u:\mathbb{C}\rightarrow \mathbb{H}^2$ satisfying $\partial u\neq 0$ with prescribed polynomial Hopf differential; there is a unique affine spherical immersion $u:\mathbb{C}\rightarrow \mathbb{R}^3$ with prescribed polynomial Pick differential. We also show that the uniqueness fails for non-polynomial entire functions with finite zeros.

math.DG↗